On linearisation and existence of preduals

Abstract

AbstractWe study the problem of existence of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space X. Then we turn to the case that $$X=\mathcal {F}(\Omega )$$ X = F ( Ω ) is a space of scalar-valued functions on a non-empty set $$\Omega $$ Ω and characterise those among them which admit a special predual, namely a strong linearisation, i.e. there are a locally convex Hausdorff space Y, a map $$\delta :\Omega \rightarrow Y$$ δ : Ω → Y and a topological isomorphism $$T:\mathcal {F}(\Omega )\rightarrow Y_{b}'$$ T : F ( Ω ) → Y b ′ such that $$T(f)\circ \delta = f$$ T ( f ) ∘ δ = f for all $$f\in \mathcal {F}(\Omega )$$ f ∈ F ( Ω ) .